MATH 605 Topics in Functional Analysis I

A graduate-level continuation of functional analysis that pushes past the basic Banach/Hilbert space machinery into the tools analysts actually use: spectra of operators, distributions as a way to differentiate things that aren't differentiable, and the Fourier transform developed rigorously across L1, L2, Schwartz space, and tempered distributions. Expect weekly problem sets, six quizzes, a midterm and final, plus a project and seminar-style presentation where you work through an advanced topic yourself from Cheney's text. It's the bridge between the abstract operator theory you saw as a first-year grad student and the PDE, harmonic analysis, and mathematical physics courses that assume you can manipulate distributions and Fourier transforms without flinching.